Newton's identities

Newton's identities

In mathematics, Newton's identities, also known as the Newton–Girard formulae, give relations between two types of symmetric polynomials, namely between power sums and elementary symmetric polynomials. Evaluated at the roots of a monic polynomial P in one variable, they allow expressing the sums of the k-th powers of all roots of P (counted with their multiplicity) in terms of the coefficients of P, without actually finding those roots. These identities were found by Isaac Newton around 1666, apparently in ignorance of earlier work (1629) by Albert Girard. They have applications in many areas of mathematics, including Galois theory, invariant theory, group theory, combinatorics, as well as further applications outside mathematics, including general relativity.

Contents

Mathematical statement

Formulation in terms of symmetric polynomials

Let x1,…, xn be variables, denote for k ≥ 1 by pk(x1,…,xn) the k-th power sum:

p_k(x_1,\ldots,x_n)=\sum\nolimits_{i=1}^nx_i^k = x_1^k+\cdots+x_n^k,

and for k ≥ 0 denote by ek(x1,…,xn) the elementary symmetric polynomial that is the sum of all distinct products of k distinct variables, so in particular

\begin{align}
e_0(x_1,\ldots,x_n) &= 1,\\
e_1(x_1,\ldots,x_n) &= x_1+x_2+\cdots+x_n,\\
e_2(x_1,\ldots,x_n) &= \textstyle\sum_{1 \leq i<j\leq n}x_ix_j,\\
e_n(x_1,\ldots,x_n) &= x_1x_2\cdots x_n,\\
e_k(x_1,\ldots,x_n) &= 0, \quad\text{for}\ k>n.\\
\end{align}

Then the Newton's identities can be stated as

 ke_k(x_1,\ldots,x_n) = \sum_{i=1}^k(-1)^{i-1} e_{k-i} (x_1,\ldots,x_n) p_i(x_1,\ldots,x_n),

valid for all k ≥ 1. Concretely, one gets for the first few values of k:

\begin{align}
 e_1(x_1,\ldots,x_n) &= p_1(x_1,\ldots,x_n),\\
 2e_2(x_1,\ldots,x_n) &= e_1(x_1,\ldots,x_n)p_1(x_1,\ldots,x_n)-p_2(x_1,\ldots,x_n),\\
 3e_3(x_1,\ldots,x_n) &= e_2(x_1,\ldots,x_n)p_1(x_1,\ldots,x_n) - e_1(x_1,\ldots,x_n)p_2(x_1,\ldots,x_n) + p_3(x_1,\ldots,x_n).\\ 
\end{align}

The form and validity of these equations do not depend on the number n of variables (although the point where the left-hand side becomes 0 does, namely after the n-th identity), which makes it possible to state them as identities in the ring of symmetric functions. In that ring one has

\begin{align}
 e_1 &= p_1,\\
 2e_2 &= e_1p_1-p_2,\\
 3e_3 &= e_2p_1 - e_1p_2 + p_3,\\
 4e_4 &= e_3p_1 - e_2p_2 + e_1p_3 - p_4,\\
\end{align}

and so on; here the left-hand sides never become zero. These equations allow to recursively express the ei in terms of the pk; to be able to do the inverse, one may rewrite them as

\begin{align}
 p_1 &= e_1,\\
 p_2 &= e_1p_1-2e_2,\\
 p_3 &= e_1p_2 - e_2p_1 + 3e_3 ,\\
 p_4 &= e_1p_3 - e_2p_2 + e_3p_1 - 4e_4, \\
     & {}\ \ \vdots
\end{align}

Application to the roots of a polynomial

Now view the xi as parameters rather than as variables, and consider the monic polynomial in t with roots x1,…,xn:

 \prod_{i=1}^n \left( t - x_i \right) = \sum_{k=0}^n (-1)^{k} a_k t^{n-k},

where the coefficients ak are given by the elementary symmetric polynomials in the roots: ak = ek(x1,…,xn). Now consider the power sums of the roots

 \psi_k = p_k(x_1,\ldots,x_n)= \sum_{i=1}^n x_i^k.

Then according to Newton's identities these can be expressed recursively in terms of the coefficients of the polynomial using

\begin{align}
  \psi_1 &= a_1,\\
  \psi_2 &= a_1 \psi_1 - 2 a_2,\\
  \psi_3 &= a_1 \psi_2 - a_2 \psi_1 + 3 a_3,\\
  \psi_4 &= a_1 \psi_3 - a_2 \psi_2 + a_3 \psi_1 - 4 a_4, \\
         & {} \  \  \vdots
  \end{align}

Application to the characteristic polynomial of a matrix

When the polynomial above is the characteristic polynomial of a matrix A, the roots xi are the eigenvalues of the matrix, counted with their algebraic multiplicity. For any positive integer k, the matrix Ak has as eigenvalues the powers xik, and each eigenvalue xi of A contributes its multiplicity to that of the eigenvalue xik of Ak. Then the coefficients of the characteristic polynomial of Ak are given by the elementary symmetric polynomials in those powers xik. In particular, the sum of the xik, which is the k-th power sum ψk of the roots of the characteristic polynomial of A, is given by its trace:

\psi_k = \operatorname{tr} ( A^k )\,.

The Newton identities now relate the traces of the powers Ak to the coefficients of the characteristic polynomial of A. Using them in reverse to express the elementary symmetric polynomials in terms of the power sums, they can be used to find the characteristic polynomial by computing only the powers Ak and their traces.

Relation with Galois theory

For a given n, the elementary symmetric polynomials ek(x1,…,xn) for k = 1,…, n form an algebraic basis for the space of symmetric polynomials in x1,…. xn: every polynomial expression in the xi that is invariant under all permutations of those variables is given by a polynomial expression in those elementary symmetric polynomials, and this expression is unique up to equivalence of polynomial expressions. This is a general fact known as the fundamental theorem of symmetric polynomials, and Newton's identities provide explicit formulae in the case of power sum symmetric polynomials. Applied to the monic polynomial \textstyle t^n+\sum_{k=1}^n (-1)^{k} a_k t^{n-k} with all coefficients ak considered as free parameters, this means that every symmetric polynomial expression S(x1,…,xn) in its roots can be expressed instead as a polynomial expression P(a1,…,an) in terms of its coefficients only, in other words without requiring knowledge of the roots. This fact also follows from general considerations in Galois theory (one views the ak as elements of a base field, the roots live in an extension field whose Galois group permutes them according to the full symmetric group, and the field fixed under all elements of the Galois group is the base field).

The Newton identities also permit expressing the elementary symmetric polynomials in terms of the power sum symmetric polynomials, showing that any symmetric polynomial can also be expressed in the power sums. In fact the first n power sums also form an algebraic basis for the space of symmetric polynomials.

Related identities

There is a number of (families of) identities that, while they should be distinguished from Newton's identities, are very closely related to them.

A variant using complete homogeneous symmetric polynomials

Denoting by hk the complete homogeneous symmetric polynomial that is the sum of all monomials of degree k, the power sum polynomials also satisfy identities similar to Newton's identities, but not involving any minus signs. Expressed as identities of in the ring of symmetric functions, they read

kh_k=\sum_{i=1}^kh_{k-i}p_i,

valid for all k ≥ 1. Contrary to Newton's identities,the left-hand sides do not become zero for large k, and the right hand sides contain ever more nonzero terms. For the first few values of k one has

\begin{align}
  h_1 &= p_1,\\
 2h_2 &= h_1p_1+p_2,\\
 3h_3 &= h_2p_1 + h_1p_2 + p_3.\\ 
\end{align}

These relations can be justified by an argument analoguous to the one by comparing coefficients in power series given above, based in this case on the generating function identity

\sum_{k=0}^\infty h_k(X_1,\ldots,X_n)t^k = \prod_{i=1}^n\frac1{1-X_it}.

The other proofs given above of Newton's identities cannot be easily adapted to prove these variants of those identities.

Expressing elementary symmetric polynomials in terms of power sums

A mentioned, Newton's identities can be used to recursively express elementary symmetric polynomials in terms of power sums. Doing so requires the introduction of integer denominators, so it can be done in the ring ΛQ of symmetric functions with rational coefficients:

\begin{alignat}2
 e_1 &= p_1,\\
 e_2 &= \textstyle\frac12p_1^2 - \frac12p_2 &&= \textstyle\frac12 ( p_1^2 - p_2 ),\\
 e_3 &= \textstyle\frac16p_1^3 - \frac12p_1 p_2 + \frac13p_3 &&= \textstyle\frac{1}{6} ( p_1^3 - 3 p_1 p_2 + 2 p_3 ),\\
 e_4 &= \textstyle\frac1{24}p_1^4 - \frac14p_1^2 p_2 + \frac18p_2^2 + \frac13p_1 p_3 - \frac14p_4 
       &&= \textstyle\frac1{24} ( p_1^4 - 6 p_1^2 p_2 + 3 p_2^2 + 8 p_1 p_3 - 6 p_4 ),\\
\end{alignat}

and so forth. Applied to a monic polynomial these formulae express the coefficients in terms of the power sums of the roots: replace each ei by ai and each pk by ψk.

Expressing complete homogeneous symmetric polynomials in terms of power sums

The analogous relations involving complete homogeneous symmetric polynomials can be similarly developed, giving equations

\begin{alignat}2
 h_1 &= p_1,\\
 h_2 &= \textstyle\frac12p_1^2 + \frac12p_2 &&= \textstyle\frac12 ( p_1^2 + p_2 ),\\
 h_3 &= \textstyle\frac16p_1^3 + \frac12p_1 p_2 + \frac13p_3 &&= \textstyle\frac{1}{6} ( p_1^3 + 3 p_1 p_2 + 2 p_3 ),\\
 h_4 &= \textstyle\frac1{24}p_1^4 + \frac14p_1^2 p_2 + \frac18p_2^2 + \frac13p_1 p_3 + \frac14p_4 
       &&= \textstyle\frac1{24} ( p_1^4 + 6 p_1^2 p_2 + 3 p_2^2 + 8 p_1 p_3 + 6 p_4 ),\\
\end{alignat}

and so forth, in which there are only plus signs. These expressions correspond exactly to the cycle index polynomials of the symmetric groups, if one interprets the power sums pi as indeterminates: the coefficient in the expression for hk of any monomial p1m1p2m2plml is equal to the fraction of all permutations of k that have m1 fixed points, m2 cycles of length 2, …, and ml cycles of length l. Explicitly, this coefficient can be written as 1 / N where N=\Pi_{i=1}^l(m_i!\,i^{m_i}); this N is the number permutations commuting with any given permutation π of the given cycle type. The expressions for the elementary symmetric functions have coefficients with the same absolute value, but a sign equal to the sign of π, namely (−1)m2+m4+….

Expressing power sums in terms of elementary symmetric polynomials

One may also use Newton's identities to express power sums in terms of symmetric polynomials, which does not introduce denominators:

\begin{align}
 p_1 &= e_1,\\
 p_2 &= e_1^2 - 2 e_2,\\
 p_3 &= e_1^3 - 3 e_2 e_1 + 3 e_3,\\
 p_4 &= e_1^4 - 4 e_2 e_1^2 + 4 e_3 e_1 + 2 e_2^2 - 4 e_4,\\
 p_5 &= e_1^5 - 5 e_2 e_1^3 + 5 e_2^2 e_1 + 5 e_3 e_1^2 - 5 e_3e_2 - 5 e_4 e_1 + 5 e_5,\\
 p_6 &= e_1^6 - 6 e_2 e_1^4 + 9 e_2^2 e_1^2 + 6 e_3 e_1^3 - 2 e_2^3 - 12 e_3 e_2 e_1 - 6 e_4 e_1^2 + 3 e_3^2 + 6 e_4 e_2 + 6 e_1 e_5 - 6e_6,\\
 \end{align}

giving ever longer expressions that do not seem to follow any simple pattern. By consideration of the relations used to obtain these expressions, it can however be seen that the coefficient of some monomial M=\Pi_{i=1}^l e_i^{m_i} in the expression for pk has the same sign as the coefficient of the corresponding product \Pi_{i=1}^l p_i^{m_i} in the expression for ek described above, namely the sign (−1)m2+m4+…. Furthermore the absolute value of the coefficient of M is the sum, over all distinct sequences of elementary symmetric functions whose product is M, of the index of the last one in the sequence: for instance the coefficient of e_1^5e_3e_4^3 in the expression for p20 will be (-1)^{0+3}(280\times1+56\times3+168\times4)=-1120, since of all distinct orderings of the five factors e1, one factor e3 and three factors e4, there are 280 that end with e1, 56 that end with e3, and 168 that end with e4.

Expressing power sums in terms of complete homogeneous symmetric polynomials

Finally one may use the variant identities involving complete homogeneous symmetric polynomials similarly to express power sums in term of them:

\begin{align}
 p_1 &= + h_1,\\
 p_2 &= - h_1^2 + 2 h_2,\\
 p_3 &= + h_1^3 - 3 h_2 h_1 + 3 h_3,\\
 p_4 &= - h_1^4 + 4 h_2 h_1^2 - 4 h_3 h_1 - 2 h_2^2 + 4 h_4,\\
 p_5 &= + h_1^5 - 5 h_2 h_1^3 + 5 h_2^2 h_1 + 5 h_3 h_1^2 - 5 h_3h_2 - 5 h_4 h_1 + 5 h_5,\\
 p_6 &= - h_1^6 + 6 h_2 h_1^4 - 9 h_2^2 h_1^2 - 6 h_3 h_1^3 + 2 h_2^3 + 12 h_3 h_2 h_1 + 6 h_4 h_1^2 - 3 h_3^2 - 6 h_4 h_2 - 6 h_1 h_5 + 6h_6,\\
\end{align}

and so on. Apart from the replacement of each ei by the corresponding hi, the only change with respect to the previous family of identities is in the signs of the terms, which in this case depend just on the number of factors present: the sign of the monomial \Pi_{i=1}^l h_i^{m_i} is −(−1)m1+m2+m3+…. In particular the above description of the absolute value of the coefficients applies here as well.

Expressions as determinants

One can obtain explicit formulas for the above expressions in the form of determinants, by considering the first n of Newton's identities (or it counterparts for the complete homogeneous polynomials) as linear equations in which the elementary symmetric functions are known and the power sums are unknowns (or vice versa), and apply Cramer's rule to find the solution for the final unknown. For instance taking Newton's identities in the form

\begin{align}
 e_1 &= 1p_1,\\
 2e_2 &= e_1p_1-1p_2,\\
 3e_3 &= e_2p_1 - e_1p_2 + 1p_3,\\
 \vdots &= \vdots \\
 ne_n &= e_{n-1}p_1 - e_{n-2} p_2 + \cdots +(-1)^ne_1p_{n-1}+(-1)^{n-1}p_n\\
\end{align}

we consider p1, p2, p3, ..., ( − 1)npn − 1 and pn as unknowns, and solve for the final one, giving

p_n=\frac{\begin{vmatrix}1 & 0 & \cdots && e_1 \\ e_1 & 1  & 0 & \cdots & 2e_2 \\  e_2 & e_1 & 1&  & 3e_3 \\ \vdots&&\ddots&\ddots&\vdots
\\ e_{n-1} & \cdots & e_2 & e_1 & ne_n \end{vmatrix}}{\begin{vmatrix}1 & 0 & \cdots & \\ e_1 & 1  & 0 & \cdots  \\  e_2 & e_1 & 1&   \\ \vdots&&\ddots&\ddots
\\ e_{n-1} & \cdots & e_2 & e_1 & (-1)^{n-1} \end{vmatrix}}
=\frac{\begin{vmatrix}1 & 0 & \cdots && e_1 \\ e_1 & 1  & 0 & \cdots & 2e_2 \\  e_2 & e_1 & 1& & 3e_3 \\ \vdots&&\ddots&\ddots&\vdots
\\ e_{n-1} & \cdots & e_2 & e_1 & ne_n \end{vmatrix}}{(-1)^{n-1}}
 =\begin{vmatrix}e_1 & 1 & 0 & \cdots\\ 2e_2 & e_1 & 1 & 0 & \cdots\\ 3e_3 & e_2 & e_1 & 1 \\ \vdots &&& \ddots & \ddots  
\\ ne_n & e_{n-1} & \cdots & & e_1 \end{vmatrix}.

Solving for en instead of for pn is similar, as the analogous computations for the complete homogeneous symmetric polynomials; in each case the details are slightly messier than the final results, which are (Macdonald 1979, p. 20):

e_n=\frac1{n!}
\begin{vmatrix}p_1 & 1 & 0 & \cdots\\ p_2 & p_1 & 2 & 0 & \cdots  \\ \vdots&& \ddots & \ddots \\ p_{n-1} & p_{n-2} & \cdots & p_1 & n-1 \\ p_n & p_{n-1} & \cdots & p_2 & p_1
\end{vmatrix},
\qquad p_n=(-1)^{n-1}
\begin{vmatrix}h_1 & 1 & 0 & \cdots\\ 2h_2 & h_1 & 1  & 0 & \cdots\\ 3h_3 & h_2 & h_1 & 1 \\ \vdots &&& \ddots & \ddots  
\\ nh_n & h_{n-1} & \cdots & & h_1
\end{vmatrix},
\qquad h_n=\frac1{n!}
\begin{vmatrix}p_1 & -1 & 0 & \cdots\\ p_2 & p_1 & -2 & 0 & \cdots \\ \vdots&& \ddots & \ddots \\ p_{n-1} & p_{n-2} & \cdots & p_1 & 1-n \\ p_n & p_{n-1} & \cdots & p_2 & p_1 \end{vmatrix}.

Note that the use of determinants makes that formula for hn has additional minus signs with respect to the one for en, while the situation for the expanded form given earlier is opposite. As remarked in (Littelwood 1950, p. 84) one can alternatively obtain the formula for hn by taking the permanent of the matrix for en instead of the determinant, and more generally an expression for any Schur polynomial can be obtained by taking the corresponding immanant of this matrix.

Derivation of the identities

Each of Newton's identities can easily be checked by elementary algebra; however, their validity in general needs a proof. Here are some possible derivations

From the special case n = k

One can obtain the k-th Newton identity in k variables by substitution into

 \prod_{i=1}^k (t - x_i) = \sum_{i=0}^k (-1)^{k-i} e_{k-i}(x_1,\ldots,x_k)t^i

as follows. Substituting xj for t gives

0= \sum_{i=0}^k (-1)^{k-i} e_{k-i}(x_1,\ldots,x_k){x_j}^i \quad\text{for }1\leq j\leq k

Summing over all j gives

0= (-1)^kke_k(x_1,\ldots,x_k)+\sum_{i=1}^k(-1)^{k-i} e_{k-i}(x_1,\ldots,x_k)p_i(x_1,\ldots,x_k),

where the terms for i = 0 were taken out of the sum because p0 is (usually) not defined. This equation immediately gives the k-th Newton identity in k variables. Since this is an identity of symmetric polynomials (homogeneous) of degree k, its validity for any number of variables follows from its validity for k variables. Concretely, the identities in n < k variables can be deduced by setting k − n variables to zero. The k-th Newton identity in n > k variables contains more terms on both sides of the equation than the one in k variables, but its validity will be assured if the coefficients of any monomial match. Because no individual monomial involves more than k of the variables, the monomial will survive the substitution of zero for some set of n − k (other) variables, after which the equality of coefficients is one that arises in the k-th Newton identity in k (suitably chosen) variables.

Comparing coefficients in series

A derivation can be given by formal manipulations based on the basic relation

\prod_{i=1}^n (t - x_i) = \sum_{k=0}^n (-1)^{k} a_k t^{n-k}

linking roots and coefficients of a monic polynomial. However, to facilitate the manipulations one first "reverses the polynomials" by substituting 1/t for t and then multiplying both sides by tn to remove negative powers of t, giving

\textstyle\prod_{i=1}^n (1- x_it) = \sum_{k=0}^n (-1)^{k} a_k t^k.

Swapping sides and expressing the ai as the elementary symmetric polynomials they stand for gives the identity

\sum_{k=0}^n (-1)^{k} e_k(x_1,\ldots,x_n) t^k=\prod_{i=1}^n (1- x_it).

One differentiates both sides with respect to t, and then (for convenience) multiplies by t, to obtain

\begin{align}
& {} \qquad \sum_{k=0}^n (-1)^{k}k e_k(x_1,\ldots,x_n) t^k \\
&=  t \sum_{i=1}^n \left((-x_i) \prod\nolimits_{j\neq i} (1- x_jt)\right)\\
 &=  -\left(\sum_{i=1}^n \frac{x_it}{1-x_it}\right) \prod\nolimits_{j=1}^n (1- x_jt)\\
 &= -\left(\sum_{i=1}^n \sum_{j=1}^\infty(x_it)^j\right) \left(\sum_{\ell=0}^n (-1)^\ell e_\ell(x_1,\ldots,x_n) t^\ell\right)\\
 &= \left(\sum_{j=1}^\infty p_j(x_1,\ldots,x_n)t^j\right) \left(\sum_{\ell=0}^n (-1)^{\ell-1} e_\ell(x_1,\ldots,x_n) t^\ell\right),\\
\end{align}

where the polynomial on the right hand side was first rewritten as a rational function in order to be able to factor out a product from of the summation, then the fraction in the summand was developed as a series in t, and finally the coefficient of each t j was collected, giving a power sum. (The series in t is a formal power series, but may alternatively be thought of as a series expansion for t sufficiently close to 0, for those more comfortable with that; in fact one is not interested in the function here, but only in the coefficients of the series.) Comparing coefficients of tk on both sides one obtains

(-1)^{k}k e_k(x_1,\ldots,x_n) = \sum_{j=1}^k (-1)^{k-j-1} p_j(x_1,\ldots,x_n)e_{k-j}(x_1,\ldots,x_n),

which gives the k-th Newton identity.

As a telescopic sum of symmetric function identities

The following derivation, given essentially in (Mead, 1992), is formulated in the ring of symmetric functions for clarity (all identities are independent of the number of variables). Fix some k > 0, and define the symmetric function r(i) for 2 ≤ i ≤ k as the sum of all distinct monomials of degree k obtained by multiplying one variable raised to the power i with k − i distinct other variables (this is the monomial symmetric function mγ where γ is a hook shape (i,1,1,…1)). In particular r(k) = pk; for r(1) the description would amount to that of ek, but this case was excluded since here monomials no longer have any distinguished variable. All products pieki can be expressed in terms of the r(j) with the first and last case being somewhat special. One has

p_ie_{k-i}=r(i)+r(i+1)\quad\text{for }1<i<k

since each product of terms on the left involving distinct variables contributes to r(i), while those where the variable from pi already occurs among the variables of the term from eki contributes to r(i + 1), and all terms on the right are so obtained exactly once. For i = k one multiplies by e0 = 1, giving trivially

p_ke_0=p_k=r(k)\,.

Finally the product p1ek−1 for i = 1 gives contributions to r(i + 1) = r(2) like for other values i < k, but the remaining contributions produce k times each monomial of ek, since any on of the variables may come from the factor p1; thus

p_1e_{k-1}=ke_k+r(2)\,.

The k-th Newton identity is now obtained by taking the alternating sum of these equations, in which all terms of the form r(i) cancel out.

See also

References

  • Tignol, Jean-Pierre (2001). Galois' theory of algebraic equations. Singapore: World Scientific. ISBN 978-981-02-4541-2. 
  • Bergeron, F.; Labelle, G.; and Leroux, P. (1998). Combinatorial species and tree-like structures. Cambridge: Cambridge University Press. ISBN 978-0-521-57323-8. 
  • Cameron, Peter J. (1999). Permutation Groups. Cambridge: Cambridge University Press. ISBN 978-0-521-65378-7. 
  • Cox, David; Little, John, and O'Shea, Donal (1992). Ideals, Varieties, and Algorithms. New York: Springer-Verlag. ISBN 978-0-387-97847-5. 
  • Eppstein, D.; Goodrich, M. T. (2007). "Space-efficient straggler identification in round-trip data streams via Newton's identities and invertible Bloom filters". Algorithms and Data Structures, 10th International Workshop, WADS 2007. Springer-Verlag, Lecture Notes in Computer Science 4619. pp. 637–648.  arXiv:0704.3313
  • Littlewood, D. E. (1950). The theory of group characters and matrix representations of groups. Oxford: Oxford University Press. viii+310. ISBN 0-8218-4067-3. 
  • Macdonald, I. G. (1979). Symmetric functions and Hall polynomials. Oxford Mathematical Monographs. Oxford: The Clarendon Press, Oxford University Press. viii+180. ISBN 0-19-853530-9. MR84g:05003. 
  • Macdonald, I. G. (1995). Symmetric functions and Hall polynomials. Oxford Mathematical Monographs (Second ed.). New York: Oxford Science Publications. The Clarendon Press, Oxford University Press. p. x+475. ISBN 0-19-853489-2. MR96h:05207. 
  • Mead, D.G. (1992-10). "Newton's Identities". The American Mathematical Monthly (Mathematical Association of America) 99 (8): 749–751. doi:10.2307/2324242. JSTOR 2324242. 
  • Stanley, Richard P. (1999). Enumerative Combinatorics, Vol. 2. Cambridge University Press. ISBN 0-521-56069-1 (hardback), ISBN 0-521-78987-7 (paperback). 
  • Sturmfels, Bernd (1992). Algorithms in Invariant Theory. New York: Springer-Verlag. ISBN 978-0-387-82445-1. 
  • Tucker, Alan (1980). Applied Combinatorics (5/e ed.). New York: Wiley. ISBN 978-0-471-73507-6. 

External links


Wikimedia Foundation. 2010.

Игры ⚽ Поможем сделать НИР

Look at other dictionaries:

  • Newton polygon — In mathematics, the Newton polygon is a tool for understanding the behaviour of polynomials over local fields. In the original case, the local field of interest was the field of formal Laurent series in the indeterminate X, i.e. the field of… …   Wikipedia

  • Isaac Newton — Sir Isaac Newton …   Wikipedia

  • Identidades de Newton — En Matemática, las identidades de Newton, también conocidas como las fórmulas Newton Girard, son dos maneras diferentes de describir la raíz de un polinomio. Concretamente, relacionan las sumas de potencias con los polinomios simétricos… …   Wikipedia Español

  • Adam Zachary Newton — is widely cited on the subject of how readers engage with text. [Long, Elizabeth. Book Clubs: Women and the Uses of Reading in Everyday Life . Chicago: University of Chicago Press, 2003. ISBN 0226492613] [Karnicky, J. (2007). Contemporary fiction …   Wikipedia

  • Symmetric polynomial — This article is about individual symmetric polynomials. For the ring of symmetric polynomials, see ring of symmetric functions. In mathematics, a symmetric polynomial is a polynomial P(X1, X2, …, Xn) in n variables, such that if any of the… …   Wikipedia

  • Power sum symmetric polynomial — In mathematics, specifically in commutative algebra, the power sum symmetric polynomials are a type of basic building block for symmetric polynomials, in the sense that every symmetric polynomial with rational coefficients can be expressed as a… …   Wikipedia

  • Cayley–Hamilton theorem — In linear algebra, the Cayley–Hamilton theorem (named after the mathematicians Arthur Cayley and William Hamilton) states that every square matrix over the real or complex field satisfies its own characteristic equation.More precisely; if A is… …   Wikipedia

  • Splitting circle method — In mathematics, the splitting circle method is a numerical algorithm for the numerical factorization of a polynomial and, ultimately, for finding its complex roots. It was introduced by Arnold Schönhage in his 1982 paper The fundamental theorem… …   Wikipedia

  • Determinant — This article is about determinants in mathematics. For determinants in epidemiology, see Risk factor. In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific… …   Wikipedia

  • Adams operation — In mathematics, an Adams operation:ψ k is a cohomology operation in topological K theory, or any allied operation in algebraic K theory or other types of algebraic construction, defined on a pattern introduced by Frank Adams. The basic idea is to …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”